This homework gives you some experience at manipulating models and data in the frequency domain. It should be turned in as an Rmd file on Canvas. There will be no class on Tuesday 2/18. The midterm exam is in class on Thursday 2/20.
Question 4.1.
A. Compute and plot the spectral density function of a stationary AR2 model, \[ X_n = 1.5 X_{n-1} - 0.8X_{n-2} + \epsilon_n,\] where \(\{\epsilon_n\}\) is white noise with \(\mathrm{Var}(\epsilon_n)=\sigma^2\). You can use software to do this, or carry out some computations analytically. It is up to you how much (or little) algebra you choose to work through, but please explain fully how you carried out your calculation. Also, plot the autocovariance function.
B. Compute and plot the spectral density function of an MA(2) moving mean, \[ X_n = \epsilon_{n-2} + \epsilon_{n-1}+\epsilon_n,\] where \(\{\epsilon_n\}\) is white noise with \(\mathrm{Var}(\epsilon_n)=\sigma^2\). As in part (A), you can use software to do this or carry out some computations analytically. Also, plot the autocovariance function.
C. Comment briefly on what you find in parts A and B.
Question 4.2. Present an estimated spectral density of the sunspot time series in sunspots.txt. Comment on your choice of estimator. Comment on the resulting estimate. These data, as well as some background on the historical and current interest in sunspot activity, are described at [http://www.sidc.be/silso/datafiles].
Question 4.3. Explain which parts of your responses above made use of a source, meaning anything or anyone you consulted (including classmates or office hours) to help you write or check your answers. All sources are permitted, but failure to attribute material from a source is unethical. See the syllabus for additional information on grading.